AFAIK, there isn't an oboe pitched an octave above the common oboe in C. There are, however, oboe musettes in Eb (usually), and sometimes in F. Heckel also made piccolo heckelphones (much like an oboe) pitched in Eb and F above the oboe. I have pictures of the heckelphone family at http://www.contrabass.com/pages/heckel.html.

Are you sure it is a piccolo oboe, and I mean the name of it? I'm sure there is a higher-pitched one, but I don't know if it would be called piccolo-oboe. Is piccolo used exclusively for flutes, or is it for any instrument pitched a full octave above the regular tone? Would it just be called a soprano oboe? I know nothing whatsoever about oboes, I am going from general band knowledge.

There is, in fact a piccolo oboe. If you're very familiar with Oboes you'll surely have heard of F. Loree (the premiere maker of oboes). This company does manufacture a piccolo oboe by that name. For a photo and some info go to their website http://www.loree-paris.com/index-gb.htm

Correction. The Marigaux site now states that their musette is in Eb. I have talked to their US rep in detail in the past and I am not sure they have ever made a musette. They state that it comes with "bocals". There are no bocals needed for a musette. Until they up dated their website they stated it was in D. I am sure that my Patricola is in Eb and that Loree is in F. As for Heckel. They have not made a picciolo instrument in many years. Jennifer Paull of Amoris has written music for the musette in F. Her group how ever performed it with my musette and transposed the score.

There was the pastoral oboe above the musette that was pitched in G or Ab made by prominent French makers in the 19th Century (eg. Triebert and some others), so pretty much a more refined version of a bombarde.